I inherited a 3/8" square notched trowel and a 3/16" v-notched trowel.
Which is the correct for laying down a coat of thin-set prior to attaching 6" x 24" tile to a wall?
When is the correct time to use either trowel?
You can use either. If your walls are perfectly flat and the tile is too then 1/4 will be fine.
What would I normally do with 6x24? I would use a 1/4" trowel but backbutter all of the pieces. The tiles need to be pushed firmly in place too. My main concern with this type of tile is the bowing in the middle.
Most tile manufacturers will have a specific trowel recommendation for their tile. It may not be either of the ones you own. In that case you'll buy another one. In any case, it is normally clearly stated in the installation instructions for the tile. The size and shape of the notches affects both the total amount of thinset (so a 3/8 X 3/8 X 3/8 notched trowel lays down an amount of thinset that would be 3/16" thick if flattened completely) and the way that the ridged thinset interacts with the tile as it is placed (V's have a bit more give than square-notches as the tile is embedded.)
Read and follow the tile manufacturer's installation instructions.